Topic 7: Future Internet Architectures...SCH-MGMT 825 Management Science Seminar Variational Inequalities, Networks, and Game Theory Spring 2014 ... next-generation Internet. Professor

Topic 7: Future Internet Architectures

Professor Anna Nagurney

John F. Smith Memorial Professorand

Director – Virtual Center for SupernetworksIsenberg School of Management

University of MassachusettsAmherst, Massachusetts 01003

SCH-MGMT 825Management Science Seminar

Variational Inequalities, Networks, and Game TheorySpring 2014c©Anna Nagurney 2014

Professor Anna Nagurney SCH-MGMT 825 Management Science Seminar

Support for the research on the project: CollaborativeResearch: Network Innovation Through Choice has beenprovided by:

This support is gratefully acknowledged.


Outline

I Background and Motivation

I Envisioning a New Kind of Internet – ChoiceNet

I Methodologies for Formulation, Analysis, andComputations

I The Game Theory Model

I Numerical Examples

I Financial Networks

I Summary and Conclusions


Background and Motivation


The Internet

• The Internet has transformed the ways in which individuals,groups, and organizations communicate, obtain information,access entertainment, and conduct their economic and socialactivities.


The Internet

In 2012, there were over 2.4 billion users


The Internet

• Many users, if not the majority, are unaware of theeconomics underlying the provision of various Internetservices.

• Although the technology associated with the existingInternet is rather well-understood, the economics of theassociated services have been less studied.

• Modeling and computational frameworks that capturethe competitive behavior of decision-makers rangingfrom service providers to network providers are still intheir infancy. This may be due, in part, to unawareness ofappropriate methodological frameworks.


The Internet





The Internet





Other Network Systems Behave Like the Internet

Transportation networks, electric power networks, supplychains, and even multitiered financial networks!


Evolution of the Internet


The Existing Internet

Much of the Internet’s success comes from its ability tosupport a wide range of service at the edge of thenetwork.

However, the Internet offers little choice of serviceinside the network.

It is widely agreed that this limitation inhibits the developmentand deployment of new networking services, protocols, securitydesigns, management frameworks, and other components thatare essential to support the increasingly diverse systems,applications, and communication paradigms of thenext-generation Internet.












Envisioning a New Kind of Internet – ChoiceNet


Envisioning a New Kind of Internet

We are one of five teams funded byNSF as part of the Future InternetArchitecture (FIA) project.

Our project is: Network InnovationThrough Choice and the envisionedarchitecture is ChoiceNet.

Team:

University of Kentucky: Jim Griffioen, KenCalvertNorth Carolina State University:Rudra Dutta, George RouskasRENCI/UNC: Ilia BaldineUniversity of Massachusetts Amherst:Tilman Wolf, Anna Nagurney


USA NSF Future Internet Architecture (FIA)

Projects

• Named Data Networking (NDN) – UCLA (lead) –Content-centric, focus on “what” not “where”

• MobilityFirst – Rutgers University (lead) – Cellularconvergence (4-5B devices) interconnected vehicles

• NEBULA – University of Pennsylvania (lead) – Reliable,high-speed core interconnecting data centers

• eXpressive Internet Architecture (XIA) – Carnegie MellonUniversity (lead) – Rich set of communication entities asnetwork principals

• ChoiceNet – University of Massachusetts Amherst (lead) –project started September 2011; assigned FIA status in 2012.


Some of Our Publications on This Project

[1] Nagurney, A., Li, D., 2013. A dynamic network oligopolymodel with transportation costs, product differentiation, andquality competition. Computational Economics, in press.

[2] Nagurney, A., Li, D., Wolf, T., Saberi, S., 2013. A networkeconomic game theory model of a service-oriented Internetwith choices and quality competition. Netnomics 14(1-2),1-25.

[3] Rouskas, G. N., Baldine, I., Calvert, K., Dutta, R.,Griffioen,J., Nagurney, A., Wolf, T., 2013. ChoiceNet:Network innovation through choice. In Proceedings of the17th Conference on Optical Network Design and Modeling(ONDM 2013), April 16-19, Brest, France. (Invited paper).


Some of Our Publications on This Project

[4] Wolf, T., Griffioen, J., Calvert, K., Dutta, R., Rouskas, G.,Baldine, I., Nagurney, A., 2012. Choice as a principle innetwork architecture. In Proceedings of ACM SIGCOMM2012, Helsinki, Finland, August 13-17.

[5] Wolf, T., Zink, M., Nagurney, A., 2013. The cyber-physicalmarketplace: A framework for large-scale horizontal integrationin distributed cyber-physical systems. In Proceedings of TheThird International Workshop on Cyber-Physical NetworkingSystems, Philadelphia, PA, July 11-13.

[6] Wolf, T., Griffioen, J., Calvert, K., Dutta, R., Rouskas, G.,Baldine, I., Nagurney, A., 2013. ChoiceNet: Toward anEconomy Plane for the Internet, December.


Network Economics Conundrums

New architectures are focusing on networkingtechnology, and not on economic interactions. Also,they lack in mechanisms to introduce competition andmarket forces.

In addition, existing economic models cannot bedeployed in today’s Internet: no mechanisms in order tocreate and discover contracts with any provider and to do soon short-time scales, and time-scales of different lengths.


ChoiceNet Goals

• Expose choices throughout the network– Network is no longer a “black box”

• Interactions between technological alternatives andrelationships – Introduction of a dynamic “economy plane”– Money as a driver to overcome inertia by providers– Market forces can play out within the network itself

• Services are at the core of ChoiceNet – “everything isa service”– Services provide a benefit but entail a cost– Services are created, composed, sold, verified, etc.


ChoiceNet Principles

Competition Drives Innovation!

Services are at core of ChoiceNet(“everything is a service”)

Services provide a benefit, have acost Services are created, composed,sold, verified, etc.

“Encourage alternatives” Providebuilding blocks for different types ofservices

“Know what happened” Ability toevaluate services

“Vote with your wallet” Rewardgood services!


































ChoiceNet Architecture


Entities in ChoiceNet

• ChoiceNet enables the composition of services andeconomic relationships– Economy plane: customer-provider relationships– Use plane: client-service relationships– Positive feature is the ability to reflect real-worldrelationships.


Entities in ChoiceNet


Provider Ecosystem

• Incentives for participation?– Everyone can be rewarded (host, verifier, author, integrator)– Innovative and good services get rewarded

• Payments among actors to sustain viability– Economy plane distributes value (i.e., money)

• Same commercial entities as today?– Similar providers, but also finer-grained providers– New providers for composition and verification.


ChoiceNet Technologies (in progress)

• Economy plane– Methods for describing composing, and instantiating services– Market places for connecting customers and providers (i.e.,search for services)– IDs associated with entities

• Use plane– Verification of the economy plane contracts in use plane– Measurement services to verify offerings.


Use Cases Enabled by ChoiceNet

• ChoiceNet / economy plane enables new businessmodels in the Internet– Very dynamic economic relationships are possible– All entities get rewarded.

• Examples– Movie streaming– reading The New York Times in a coffee shop (short-termand long-term contracts)–Customers as providers.


Methodologies for Formulation, Analysis,and Computations


The Variational Inequality Problem

We utilize the theory of variational inequalities for theformulation, analysis, and solution of both centralized anddecentralized network problems.

Definition: The Variational Inequality ProblemThe finite-dimensional variational inequality problem,VI(F ,K), is to determine a vector X ∗ ∈ K, such that:

〈F (X ∗), X − X ∗〉 ≥ 0, ∀X ∈ K,

where F is a given continuous function from K to RN , K is agiven closed convex set, and 〈·, ·〉 denotes the inner product inRN .


The variational inequality problem contains, as special cases,such mathematical programming problems as:

• systems of equations,

• optimization problems,

• complementarity problems,

• and is related to the fixed point problem.

Hence, it is a natural methodology for a spectrum ofcongested network problems from centralized todecentralized ones as well as to design problems.


Geometric Interpretation of VI(F ,K) and a

Projected Dynamical System (Dupuis and

Nagurney, Nagurney and Zhang)

In particular, F (X ∗) is “orthogonal” to the feasible set K atthe point X ∗.

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Associated with a VI is a Projected Dynamical System, whichprovides a natural underlying dynamics.


To model the dynamic behavior of the Internet, we utilizeprojected dynamical systems (PDSs) advanced by Dupuis andNagurney (1993) in Annals of Operations Research and byNagurney and Zhang (1996) in our book Projected DynamicalSystems and Variational Inequalities with Applications.

Such nonclassical dynamical systems are now being used in

evolutionary games (Sandholm (2005, 2011)),

ecological predator-prey networks (Nagurney andNagurney (2011a, b)), and

even neuroscience (Girard et al. (2008)).


Background Material


The Game Theory Model


Some Features of Our Model

We build on the recent work on game theory frameworks for aservice-oriented Internet with the goal of expanding thegenerality of applicable game theory models that are alsocomputable.

By services we mean not only content, such as news,videos, music, etc., but also services associated with,for example, cloud computing.

The game theory model that we present here is inspired bythat of Zhang, Nabipay, Odlyzko, and Guerin (2010)who employed Cournot and Bertrand games to modelcompetition among service providers and amongnetwork providers, with the former competing in a Cournotmanner, and the latter in a Bertrand manner. The two typesof competition were then unified in a Stackelberg game.



Zhang et al. (2010) focused only on a two service provider,two network provider, and two user networkconfiguration along with a linear demand function toenable closed form analytical solutions. They did notcapture the quality of network provision.

Altman et al. (2011) emphasized the need for metrics forquality of service and the Internet and also provided anexcellent review of game theory models, and noted that manyof the models in the existing literature considered only one ortwo service providers.



A notable feature of our modeling approach is that itallows for composition, in that users at demand markets haveassociated demand price functions that reflect how much theyare willing to pay for the service and the network provisioncombination, as a function of service volumes and qualitylevels. Such an idea is motivated, in part, to provideconsumers with more choices (see Wolf et al. (2012)).

Our framework can be used as the foundation for thefurther disaggregation of decision-making and theinclusion of additional topological constructs, say, inexpanding the paths, which may reflect the transport ofservices at the more detailed level of expandedsequences of links.


Some More References

Our contributions fall under network economics as well ascomputational management science.

Some of the early papers on network economics and theInternet are the works of: MacKie-Mason and Varian (1995),Varian (1996), Kelly (1997), MacKnight and Bailey (1997),Kausar, Briscoe, and Crowcroft (1999), and Odlyzko (2000).

More recent contributions: Ros and Tuffin (2004), He andWalrand (2005), Shakkottai and Srikant (2006), Shen andBasar (2007), and Neely (2007).


Some More References

Lv and Rouskas (2010) focused on the modeling of Internetservice providers and the pricing of tiered network services.They provided both models and an algorithm, alongwith computational results, a contribution that is rare inthis stream of literature. They assumed that the usersare homogeneous, whereas we consider distinct demandprice functions associated with the demand markets and thecomposition of service provider services and network provision.



This part of the lectre is based on the paper, “ACournot-Nash-Bertrand Game Theory Model of aService-Oriented Internet with Price and Quality CompetitionAmong Network Transport Providers,” A. Nagurney and T.Wolf, Computational Management Science, (2013), in press.



There are m service providers, with a typical service providerdenoted by i , n network providers, which provide “transport”of the services to the demand markets, with a typical onedenoted by j , and o demand markets associated with the usersof the services and network provision. A typical demandmarket is denoted by k .

The service providers offer multiple different services such asmovies for video streaming, music for downloading, news, etc.Users can select among different service offerings (e.g., moviestreaming from service provider 1 vs. movie streaming fromservice provider 2).

Different network providers can be used for datacommunication over the Internet (i.e., “transport”) betweenthe service providers and the users.

In addition, we explicitly handle the quality level amongnetwork providers.


The Model

We allow for consumers to differentiate among the servicesprovided by the service providers.

It is assumed that the service providers compete under theCournot-Nash equilibrium concept of non-cooperative behaviorand select their service volumes (quantities).

The network providers, in turn, compete with prices a laBertrand and with quality levels.

The consumers, in turn, signal their preferences for theservices and network provision via the demand price functionsassociated with the demand markets. The demand pricefunctions are, in general, functions of the service/networkprovision combinations at all the demand markets as well asthe quality levels of network provision, since the focus here ison composition and having choices.


The Model






The Model






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Network Providers

Demand Markets

Service Providers

Figure: The Network Structure of the Cournot-Nash-BertrandModel for a Service-Oriented Internet


Table: Notation for the Model

Notation DefinitionQijk nonnegative service volume from i to k via j .

We group the {Qijk} elements into vector Q ∈ Rmno+ .

si service volume (output) produced by service provideri .We group the {si} elements into vector s ∈ Rm

+ .

dijk demand for service i transported by j to demand mar-ket k.We group the {dijk} elements into vector d ∈ Rmno .

qijk nonnegative quality level of network provider j trans-porting service i to k. We group {qijk} elements intovector q ∈ Rmno

+ .

πijk price charged by network provider j for transporting aunit of service provided by i via j to k.We group the {πijk} elements into vector π ∈ Rmno .


Table: Notation for the Model

Notation Definition

fi (s) total production cost of service provider i .

ρijk(d , q) demand price at k with service i transported via j .

cijk(Q, q) transportation cost with delivering service i via j to k.

ocijk(πijk) opportunity cost with pricing by network provider jservices from i to k.


The Behavior of the Service Providers and Their

Optimality Conditions

The service providers seek to maximize their individual profits,where the profit function for service provider i ; i = 1, . . . , m isgiven by the expression:

n∑j=1

o∑k=1

ρijk(d , q∗)Qijk − fi(s)−n∑

j=1

o∑k=1

π∗ijkQijk (1)

subject to the constraints:

si =n∑

j=1

o∑k=1

Qijk , i = 1, . . . , m, (2)

dijk = Qijk , i = 1, . . . , m; j = 1, . . . , n, (3)

Qijk ≥ 0, j = 1, . . . , n; k = 1, . . . , o. (4)


The Behavior of the Service Providers andTheir Optimality Conditions

In view of constraint (2), we can define the production costfunctions fi(Q); i = 1, . . . , m, as follows:

fi(Q) ≡ fi(s), (5)

and, in view of constraint (3), we can also define the demandprice functions ρijk(Q, q); i = 1, . . . , m; j = 1, . . . , n;k = 1, . . . , o, such that

ρijk(Q, q) ≡ ρijk(d , q). (6)

We assume that the production cost and the demand pricefunctions are continuous and continuously differentiable. Wealso assume that the production cost functions are convex andthat the demand price functions are monotonically decreasingin service volumes but increasing in the quality of networkprovision.


The Behavior of the Service Providers andTheir Optimality Conditions

Therefore, the profit maximization problem for service provideri ; i = 1, . . . , m, with its profit expression denoted by U1

i ,which also represents its utility function, with the superscript 1reflecting the first (top) tier of decision-makers in Figure 1,can be reexpressed as:

Maximize U1i (Q, q∗, π∗) =

n∑j=1

o∑k=1

ρijk(Q, q∗)Qijk − fi(Q)

−n∑

j=1

o∑k=1

π∗ijkQijk (7)

subject to: (4).




For service provider i , we group all its {Qijk} elements, whichare its strategic variables, into vector Qi . The strategicvariables of service provider i are its service transport volumes{Qi}. In view of (1) - (7), we may write the profit functions ofthe service providers as functions of the serviceprovision/transportation pattern, that is,

U1 = U1(Q, q, π), (8)

where U1 is the m-dimensional vector with components:{U1

1 , . . . , U1m}. Let K 1i denote the feasible set corresponding

to service provider i , where K 1i ≡ {Qi |Qi ≥ 0} and defineK 1 ≡

∏mi=1 K 1i .




We consider the oligopolistic market mechanism, in which them service providers supply their services in a non-cooperativefashion, each one trying to maximize its own profit.

We seek to determine a nonnegative service volume patternQ∗ for which the m service providers will be in a state ofequilibrium as defined below. In particular, Nash (1950, 1951)generalized Cournot’s concept of an equilibrium among severalplayers, in what has been come to be called a non-cooperativegame.


Definition 1: Cournot-Nash Equilibrium with ServiceDifferentiation and Network Provision Choices. Aservice volume pattern Q∗ ∈ K 1 is said to constitute aCournot-Nash equilibrium if for each service provider i ;i = 1, . . . , m:

U1i (Q∗

i , Q∗i , q

∗, π∗) ≥ U1i (Qi , Q∗

i , q∗, π∗), ∀Qi ∈ K 1i , (9)

whereQ∗

i ≡ (Q∗1 , . . . , Q

∗i−1, Q

∗i+1, . . . , Q

∗m). (10)


Theorem 1: Variational Inequality Formulations ofCournot-Nash Equilibrium. Assume that for each serviceprovider i the profit function U1

i (Q, q, π) is concave withrespect to the variables in {Qi} and is continuous andcontinuously differentiable. Then, Q∗ ∈ K 1 is a Cournot-Nashequilibrium according to Definition 1 if and only if it satisfies

−m∑

i=1

n∑j=1

o∑k=1

∂U1i (Q∗, q∗, π∗)

∂Qijk× (Qijk −Q∗

ijk) ≥ 0, ∀Q ∈ K 1,

(11)or, equivalently, Q∗ ∈ K 1 is a Cournot-Nash equilibriumservice volume pattern if and only if it satisfies the VI

m∑i=1

n∑j=1

o∑k=1

[∂ fi (Q

∗)

∂Qijk+ π∗ijk − ρijk(Q∗, q∗)−

n∑h=1

o∑l=1

∂ρihl(Q∗, q∗)

∂Qijk× Q∗ihl

]

×(Qijk − Q∗ijk) ≥ 0, ∀Q ∈ K 1. (12)


Proof: (11) follows directly from Gabay and Moulin (1980)and Dafermos and Nagurney (1987). In order to obtain (12)from (11), we note that ∀i , j , k :

−∂U1i (Q∗, q∗, π∗)

∂Qijk=

[∂ fi(Q

∗)

∂Qijk+ π∗ijk − ρijk(Q

∗, q∗)−n∑

h=1

o∑l=1


∂Qijk× Q∗

ihl

].(13)

Multiplying the expression in (13) by (Qijk − Q∗ijk) and

summing the resultant over all i , j , and k yields (12). �


The Behavior of the Network Providers and Their


The network providers also seek to maximize their individualprofits. They have as their strategic variables the prices thatthey charge for the transport of the services and the qualitylevels. The optimization problem faced by network provider j ;j = 1, . . . , n is given by

Maximize U2j (Q∗, q, π) =

m∑i=1

o∑k=1

πijkQ∗ijk −

m∑i=1

o∑k=1

cijk(Q∗, q)

−ocijk(πijk) (14)

subject to:

πijk ≥ 0, i = 1, . . . , m; k = 1, . . . , o, (15)

qijk ≥ 0, i = 1, . . . , m; k = 1, . . . , o. (16)


The Behavior of the Network Providers and Their


We group network provider j ’s prices {πijk} into the vector πj

and its quality levels {qijk} into the vector qj . We also groupthe network provider utility functions, as given in (14), intothe vector U2 as in (17):

U2 = U2(Q, q, π). (17)

Let K 2j denote the feasible set corresponding to networkprovider j , such that K 2j≡{(qj , πj)|qj ≥ 0, πj ≥ 0} andK 2 ≡

∏nj=1 K 2j .


Definition 2: Bertrand Equilibrium in Transport Pricesand Quality. A quality level pattern and transport pricepattern (q∗, π∗) ∈ K 2 is said to constitute a Bertrandequilibrium if for each network provider j ; j = 1, . . . , n:

U2j (Q∗, q∗j , q

∗j , π

∗j , π

∗j ) ≥ U2

j (Q∗, qj , q∗j , πj , π∗j ), ∀(qj , πj) ∈ K 2j ,(18)

whereq∗j ≡ (q∗1 , . . . , q

∗j−1, q

∗j+1, . . . , q

∗n), (19)

π∗j ≡ (π∗1, . . . , π∗j−1, π

∗j+1, . . . , π

∗n). (20)

According to (18), a Bertrand equilibrium is established if nonetwork provider can unilaterally improve upon its profits byselecting an alternative vector of quality levels and transportprices.


Theorem 2: Variational Inequality Formulations ofBertrand Equilibrium. Assume that for each networkprovider j the profit function U2

j (Q, q, π) is concave withrespect to the variables in {qj} and in {πj} and is continuousand continuously differentiable. Then, (q∗, π∗) ∈ K 2 is aBertrand equilibrium according to Definition 2 if and only if itsatisfies the variational inequality

−n∑

j=1

m∑i=1

o∑k=1

∂U2j (Q∗, q∗, π∗)

∂qijk× (qijk − q∗ijk)

−n∑

j=1

m∑i=1

o∑k=1

∂U2j (Q∗, q∗, π∗)

∂πijk×(πijk−π∗ijk) ≥ 0, ∀(q, π) ∈ K 2,

(21)


or, equivalently, (q∗, π∗) ∈ K 2 is a Bertrand price and qualitylevel equilibrium pattern if and only if it satisfies

n∑j=1

m∑i=1

o∑k=1

[m∑

h=1

o∑l=1

∂chjl(Q∗, q∗)

∂qijk

]× (qijk − q∗ijk)

+n∑

j=1

m∑i=1

o∑k=1

[−Q∗

ijk +∂oc(π∗ijk)

∂πijk

]×(πijk−π∗ijk) ≥ 0, ∀(q, π) ∈ K 2.

(22)Proof: Similar to the proof of Theorem 1.


The Integrated Cournot-Nash-Bertrand Equilibrium

Conditions and Variational Inequality Formulations

We are now ready to present the Cournot-Nash-Bertrandequilibrium conditions. We let K 3 ≡ K 1 × K 2 denote thefeasible set for the integrated model. We assume the sameassumptions on the functions as previously.

Definition 3: Cournot-Nash-Bertrand Equilibrium inService Differentiation, Transport Network Prices, andQuality. A service volume, quality level, and transport pricepattern (Q∗, q∗, π∗) ∈ K 3 is a Cournot-Nash-Bertrandequilibrium if it satisfies (9) and (18) simultaneously.


Theorem 3: Variational Inequality Formulations ofCournot-Nash-Bertrand Equilibrium. Under the sameassumptions as given in Theorems 1 and 2, (Q∗, q∗, π∗) ∈ K 3

is a Cournot-Nash-Bertrand equilibrium according to Definition3 if and only if it satisfies the variational inequality:

−m∑

i=1

n∑j=1

o∑k=1

∂U1i (Q∗, q∗, π∗)

∂Qijk× (Qijk − Q∗

ijk)

−n∑

j=1

m∑i=1

o∑k=1

∂U2j (Q∗, q∗, π∗)

∂qijk× (qijk − q∗ijk)

−n∑

j=1

m∑i=1

o∑k=1

∂U2j (Q∗, q∗, π∗)

∂πijk×(πijk−π∗ijk) ≥ 0, ∀(Q, q, π) ∈ K 3,

(23)


or, equivalently, the variational inequality problem:

m∑i=1

n∑j=1

o∑k=1

[∂ fi(Q

∗)

∂Qijk+ π∗ijk

−ρijk(Q∗, q∗)−

n∑h=1

o∑l=1


∂Qijk× Q∗

ihl

]× (Qijk − Q∗

ijk)

+n∑

j=1

m∑i=1

o∑k=1

[m∑

h=1

o∑l=1

∂chjl(Q∗, q∗)

∂qijk

]× (qijk − q∗ijk)

+n∑

j=1

m∑i=1

o∑k=1

[−Q∗

ijk +∂ocijk(π

∗ijk)

∂πijk

]× (πijk − π∗ijk) ≥ 0,

∀(Q, q, π) ∈ K 3. (24)


We now put variational inequality (24) into standard form:determine X ∗ ∈ K where X is a vector in RN , F (X ) is acontinuous function such that F (X ) : X 7→ K ⊂ RN , and

〈F (X ∗), X − X ∗〉 ≥ 0, ∀X ∈ K, (25)

where 〈·, ·〉 is the inner product in the N-dimensionalEuclidean space, and K is closed and convex. We define thevector X ≡ (Q, q, π) and K ≡ K 3. Also, here N = 3mno.The components of F are then given by: ∀i , j , k :

F 1ijk(X ) =

∂ fi(Q)

∂Qijk+πijk−ρijk(Q, q)−

n∑h=1

o∑l=1

∂ρihl(Q, q)

∂Qijk×Qihl ,

(26)

F 2ijk(X ) =

m∑h=1

o∑l=1

∂chjl(Q, q)

∂qijk, (27)

F 3ijk(X ) = −Qijk +

∂ocijk(πijk)

∂πijk. (28)


The Underlying Dynamics



In our framework, the rate of change of the service volumebetween a service provider i and demand market k via networkprovider j is in proportion to −F 1

ijk(X ), as long as the servicevolume Qijk is positive. Namely, when Qijk > 0,

Qijk =∂U1

i (Q, q, π)

∂Qijk, (29)

where Qijk denotes the rate of change of Qijk . However, whenQijk = 0, the nonnegativity condition (4) forces the service

volume Qijk to remain zero when∂U1

i (Q,q,π)

∂Qijk≤ 0. Hence, in this

case, we are only guaranteed of having possible increases ofthe service volume. Namely, when Qijk = 0,

Qijk = max{0, ∂U1i (Q, q, π)

∂Qijk}. (30)



We may write (29) and (30) concisely as:

Qijk =

{∂U1

i (Q,q,π)

∂Qijk, if Qijk > 0

max{0, ∂U1i Q,q,π)

∂Qijk}, if Qijk = 0.

(31)



As for the quality levels (cf. (16)), when qijk > 0, then

qijk =∂U2

j (Q, q, π)

∂qijk, (32)

where qijk denotes the rate of change of qijk ; otherwise:

qijk = max{0,∂U2

j (Q, q, π)

∂qijk}, (33)

since qijk must be nonnegative.Combining (32) and (33), we may write:

qijk =

∂U2

j (Q,q)

∂qijk, if qijk > 0

max{0, ∂U2j (Q,q)

∂qijk}, if qijk = 0.

(34)


The Underlying Dynamics and Stability Analysis

Using similar arguments as above, we can conclude that forthe network transport prices, when πijk > 0, then

πijk =∂U2

j (Q, q, π)

∂πijk, (35)

where πijk denotes the rate of change of πijk ; otherwise (cf.(15)):

πijk = max{0,∂U2

j (Q, q, π)

∂πijk}, (36)

since πijk must be nonnegative. Hence, we have that:

πijk =

∂U2

j (Q,q,π)

∂πijk, if πijk > 0

max{0, ∂U2j (Q,q,π)

∂πijk}, if πijk = 0.

(37)



Applying (31), (34), and (37) to all i = 1, . . . , m; j = 1, . . . , n,and k = 1, . . . , o, and combining the resultants, yields thefollowing pertinent ordinary differential equation (ODE) forthe adjustment processes of the service volumes, quality levels,and transport network prices, in vector form, as:

X = ΠK(X ,−F (X )), (38)

where, since K is a convex polyhedron, according to Dupuisand Nagurney (1993), ΠK(X ,−F (X )) is the projection, withrespect to K, of the vector −F (X ) at X defined as

ΠK(X ,−F (X )) = limδ→0

PK(X − δF (X ))− X

δ(39)

with PK denoting the projection map:

P(X ) = argminz∈K‖X − z‖, (40)

and where ‖ · ‖ = 〈x , x〉. Hence, F (X ) = −∇U(Q, q, π),where ∇U(Q, q, π) is the vector of marginal utilities (profits).



We cite the following theorem from Dupuis and Nagurney(1993). See also the book by Nagurney and Zhang (1996).

Theorem 4X ∗ solves the variational inequality problem (25) if and only ifit is a stationary point of the ODE (38), that is,

X = 0 = ΠK(X ∗,−F (X ∗)). (41)

This theorem demonstrates that the necessary and sufficientcondition for a pattern X ∗ = (Q∗, q∗, π∗) to be aCournot-Nash-Bertrand equilibrium, according to Definition 3,is that X ∗ = (Q∗, q∗, π∗) is a stationary point of theadjustment process defined by ODE (38), that is, X ∗ is thepoint at which X = 0.


The Algorithm

The projected dynamical system yields continuous-timeadjustment processes. However, for computational purposes, adiscrete-time algorithm, which serves as an approximation tothe continuous-time trajectories is needed.

We now recall the Euler method, which is induced by thegeneral iterative scheme of Dupuis and Nagurney (1993).Specifically, iteration τ of the Euler method (see alsoNagurney and Zhang (1996)) is given by:

X τ+1 = PK(X τ − aτF (X τ )), (42)

where PK is the projection on the feasible set K and F is thefunction that enters the variational inequality problem (19).


The Algorithm

As shown in Dupuis and Nagurney (1993) and Nagurney andZhang (1996), for convergence of the general iterative scheme,which induces the Euler method, the sequence {aτ} mustsatisfy:

∑∞τ=0 aτ = ∞, aτ > 0, aτ → 0, as τ →∞.

Specific conditions for convergence of this scheme as well asvarious applications to the solutions of other game theorymodels can be found in Nagurney, Dupuis, and Zhang (1994),Nagurney, Takayama, and Zhang (1995), Cruz (2008),Nagurney (2010), and Nagurney and Li (2012).


The Algorithm

The elegance of this procedure for the computation ofsolutions to our model (in both the dynamic and static, thatis, equilibrium, versions) can be seen in the following explicitformulae.

Explicit Formulae for the Euler Method Applied to theCournot-Nash-Bertrand Game Theory ModelWe have the following closed form expression for the servicevolumes:

Qτ+1ijk = max{0, Qτ

ijk+aτ (ρijk(Qτ , qτ )+

n∑h=1

o∑l=1

∂ρihl(Qτ , qτ )

∂Qijk×Qτ

ihl

−πτijk −

∂ fi(Qτ )

∂Qijk)}, ∀i , j , k ,


The Algorithm

and the following closed form expression for the quality levels:

qτ+1ijk = max{0, qτ

ijk + aτ (−m∑

h=1

o∑l=1

∂chjl(Qτ , qτ )

∂qijk)}, ∀i , j , k ,

with the explicit formulae for the network transport pricesbeing:

πτ+1ijk = max{0, πτ

ijk + aτ (Qτijk −

∂ocijk(πijk)

∂πijk)}, ∀i , j , k .


The Algorithm

We now provide the convergence result. The proof is directfrom Theorem 5.8 in Nagurney and Zhang (1996).

Theorem 5In the Cournot-Nash-Bertrand model for a service-orientedInternet, let F (X ) = −∇U(Q, q, π) be strongly monotone.Also, assume that F is uniformly Lipschitz continuous. Thenthere exists a unique equilibrium service volume, quality leel,and price pattern (Q∗, q∗, π∗) ∈ K and any sequencegenerated by the Euler method as given by (42) above, where{aτ} satisfies

∑∞τ=0 aτ = ∞, aτ > 0, aτ → 0, as τ →∞

converges to (Q∗, q∗, π∗).


Numerical Examples


Numerical Examples

We applied the Euler method to compute the Cournot - Nash- Bertrand equilibrium for several examples. We set {aτ} =1(1, 1

2, 1

2, 1

3, 1

3, 1

3, . . .). The convergence criterion was that the

absolute value of the difference of the iterates at twosuccessive iterations was less than or equal to 10−4. Therewere 3 service providers, 2 network providers, and 2 demandmarkets.

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Baseline Example 1

The data for this numerical example, from which we thenconstruct subsequent variants, were as follows.

The production cost functions were:

f1 = 2(Q111 +Q112 +Q121 +Q122)2 +(Q111 +Q112 +Q121 +Q122),

f2 = (Q211 + Q212 + Q221 + Q222)2 + (Q211 + Q212 + Q221 + Q222),

f3 = 3(Q311 +Q312 +Q321 +Q322)2 +(Q311 +Q312 +Q321 +Q322).


Baseline Example 1

The demand price functions were:

ρ111 = −Q111−.5Q112+q111+100, ρ112 = −2Q112−1Q111+q112+200,

ρ121 = −2Q121−.5Q111+.5q121+100, ρ122 = −3Q122−Q112+.5q122+150,

ρ211 = −1Q211−.5Q212+.3q211+100, ρ212 = −3Q212+.8q212+200,

ρ221 = −2Q221−1Q222+q221+140, ρ222 = −3Q222−Q121+q221+300,

ρ311 = −4Q311+.5q311+230, ρ312 = −2Q312−Q321+.3q312+150,

ρ321 = −3Q321−Q311+.2q321+200, ρ322 = −4Q322+.7q322+300.


Baseline Example 1

The transportation cost functions were:

c111 = q2111 − .5q111, c112 = .5q2

112 − q112, c121 = .1q2121 − q121,

c122 = q2122,

c211 = .1q2211 − q211, c212 = q2

212 − .5q212, c221 = 2q2221,

c222 = .5q2222 − q222,

c311 = q2311 − q311, c312 = .5q2

312 − q312, c321 = q2321 − q321,

c322 = 2q2322 − 2q322.


Baseline Example 1

The opportunity cost functions were:

oc111 = 2π2111, oc112 = 2π2

112, oc121 = π2121, oc122 = .5π2

122,

oc211 = π2211, oc212 = .5π2

212, oc221 = 2π2221, oc222 = 1.5π2

222,

oc311 = π2311, oc312 = 2.5π2

312, oc321 = 1.5π2321, oc322 = π2

322.

The Euler method converged in 432 iterations and yielded theapproximation to the equilibrium solution reported in the nextTable.


Table: Equilibrium Solution for the Baseline Example 1

Service Network DemandProvider i Provider j Market k Q∗

ijk q∗ijk π∗ijk1 1 1 0.00 0.25 0.001 1 2 22.67 1.00 5.671 2 1 0.00 5.00 0.001 2 2 3.24 0.00 3.242 1 1 0.00 5.00 0.002 1 2 14.53 0.25 14.532 2 1 2.24 0.00 0.562 2 2 31.97 1.00 10.663 1 1 7.55 0.50 3.773 1 2 0.00 1.00 0.003 2 1 4.18 0.50 1.393 2 2 15.80 0.50 7.90


Baseline Example 1

The profit of service provider 1 was: 2402.31 and that ofservice provider 2: 6086.77 and service provider 3: 3549.49.The profit of network provider 1 was: 184.04 and that ofnetwork provider 2 was: 241.54.

It is interesting to see that demand market 1 obtainsno services from service provider 1 since Q∗

111 and Q∗121

are equal to 0.00 and only obtains services from serviceproviders 2 and 3.Demand market 2, however, obtains services from all threeservice providers. Network provider 1 handles positive volumesof services from all service providers as does network provider2. It is also interesting to see that two of the quality levels areequal to zero.Noting that Q∗

111 = 0.00 we then constructed Variant 1 asfollows.


Baseline Example 1

The profit of service provider 1 was: 2402.31 and that ofservice provider 2: 6086.77 and service provider 3: 3549.49.The profit of network provider 1 was: 184.04 and that ofnetwork provider 2 was: 241.54.It is interesting to see that demand market 1 obtainsno services from service provider 1 since Q∗

111 and Q∗121

are equal to 0.00 and only obtains services from serviceproviders 2 and 3.

Demand market 2, however, obtains services from all threeservice providers. Network provider 1 handles positive volumesof services from all service providers as does network provider2. It is also interesting to see that two of the quality levels areequal to zero.Noting that Q∗



Baseline Example 1


111 and Q∗121

are equal to 0.00 and only obtains services from serviceproviders 2 and 3.Demand market 2, however, obtains services from all threeservice providers. Network provider 1 handles positive volumesof services from all service providers as does network provider2. It is also interesting to see that two of the quality levels areequal to zero.

Noting that Q∗111 = 0.00 we then constructed Variant 1 as

follows.


Baseline Example 1


111 and Q∗121

are equal to 0.00 and only obtains services from serviceproviders 2 and 3.Demand market 2, however, obtains services from all threeservice providers. Network provider 1 handles positive volumesof services from all service providers as does network provider2. It is also interesting to see that two of the quality levels areequal to zero.Noting that Q∗



Example 2: Variant 1 of Example 1

We explored the effects of a change in the price function ρ111

since, in Example 1, Q∗111 = 0.00. Such a change in a price

function could occur, for example, through enhancedmarketing.

Specifically, we sought to determine the change in theequilibrium pattern if the consumers at demand market 1 arewilling to pay more for the services of the service provider 1and network provider 1 combination.

The new demand price function was:

ρ111(Q, q) = −Q111 − .5Q112 + q111 + 200,

with the remainder of the data as in Example 1. The newcomputed solution is reported in the next Table. Thealgorithm converged in 431 iterations.


Table: Equilibrium Solution for Example 2: Variant 1 of Example 1


ijk q∗ijk π∗ijk1 1 1 25.40 0.25 6.351 1 2 8.67 1.00 2.171 2 1 0.00 4.45 0.001 2 2 0.37 0.00 0.372 1 1 0.00 4.45 0.002 1 2 14.52 0.25 14.532 2 1 2.24 0.00 0.562 2 2 31.97 1.00 10.663 1 1 7.55 0.50 3.773 1 2 0.00 1.00 0.003 2 1 4.18 0.50 1.393 2 2 15.80 0.50 7.90



The profit of service provider 1 was now: 3168.18. The profitsof the other two service providers remained as in Example 1.The profit of network provider 1 was now: 209.85 and that ofnetwork provider 2 was: 236.35.

Hence, both service provider 1 and network provider 1 hadhigher profits than in Example 1 and the service volume Q∗

111

increased from 0.00 to 25.40. There was a reduction in servicevolume Q∗

112 and in Q∗122.



The profit of service provider 1 was now: 3168.18. The profitsof the other two service providers remained as in Example 1.The profit of network provider 1 was now: 209.85 and that ofnetwork provider 2 was: 236.35.

Hence, both service provider 1 and network provider 1 hadhigher profits than in Example 1 and the service volume Q∗

111

increased from 0.00 to 25.40. There was a reduction in servicevolume Q∗

112 and in Q∗122.



In the final example, we returned to Example 1 and modifiedall of the transportation cost functions to include an additionalterm: Qijkqijk to reflect that cost could depend on bothcongestion level and on quality of transport.

The solution obtained via the Euler method for this example isgiven in the next Table. The Euler method required 705iterations for convergence.


Table: Equilibrium Solution for Example 3: Variant 2 of Example 1


ijk q∗ijk π∗ijk1 1 1 0.00 0.25 0.001 1 2 22.52 0.00 5.631 2 1 0.00 4.98 0.001 2 2 3.31 0.00 3.312 1 1 0.00 4.99 0.002 1 2 14.52 0.00 14.522 2 1 2.31 0.00 0.582 2 2 31.84 0.00 10.613 1 1 7.53 0.00 3.773 1 2 0.00 1.00 0.003 2 1 4.19 0.00 1.403 2 2 15.77 0.00 7.89



The profit of service provider 1 was now: 2380.87. The profitof service provider 2 was: 6053.76 and that of service provider3 was: 3541.93. The profit of network provider 1 was now:181.89 and that of network provider 2 was: 237.21.

Observe that, in this, as in the previous two examples, ifQ∗

ijk = 0, then the price π∗ijk = 0, which is reasonable.

It is interesting to note that, in this example, the inclusion ofan additional term Qijkqijk to each transportation cost functioncijk , with the remainder of the data as in Example 1, results ina decrease in the quality levels in eight out of the twelvecomputed equilibrium variable values, with the other qualityvalues remaining unchanged.














Having an effective modeling and computationalframework allows one to explore the effects of changesin the underlying functions on the equilibrium patternto gain insights that may not be apparent from smallerscale, analytical solutions.


Related Project

Parallel to our NSF Choicenet project we have been workingon a project funded by the Advanced Cyber Security Center,entitled “Cybersecurity Risk Analysis and Security InvestmentOptimization.”

The PI on this project is W. Burleson, with Co-PIs: A.Nagurney, M. Sherman, S. Solak, and C. Misra, all at UMassAmherst.

This project aims to assess the vulnerability of financialnetworks with a focus on cybersecurity.


Financial Networks


Financial Networks

The study of financial networks dates to the 1750s whenQuesnay (1758), in his Tableau Economique, conceptualizedthe circular flow of financial funds in an economy as a network.


Financial Networks

The advances in information technology and globalization havefurther shaped today’s financial world into a complex network,which is characterized by distinct sectors, the proliferation ofnew financial instruments, and with increasing internationaldiversification of portfolios.


Financial Networks

As pointed out by Sheffi (2005) in his book, The ResilientEnterprise, one of the main characteristics of disruptions innetworks is “the seemingly unrelated consequences andvulnerabilities stemming from global connectivity.”


Financial Networks

In 2008 and 2009, the world reeled from the effects of thefinancial credit crisis; leading financial services and banksclosed (including the investment bank Lehman Brothers),others merged, and the financial landscape was changed forforever.

The domino effect of the U.S. economic troublesrippled through overseas markets and pushed countriessuch as Iceland to the verge of bankruptcy.


Financial Networks

In 2008 and 2009, the world reeled from the effects of thefinancial credit crisis; leading financial services and banksclosed (including the investment bank Lehman Brothers),others merged, and the financial landscape was changed forforever.

The domino effect of the U.S. economic troublesrippled through overseas markets and pushed countriessuch as Iceland to the verge of bankruptcy.


Financial Networks

It is crucial for the decision-makers in financial systems(managers, executives, and regulators) to be able to identifya financial network’s vulnerable components to protectthe functionality of the network.

Financial networks, as extremely importantinfrastructure networks, have a great impact on theglobal economy, and their study has recently alsoattracted attention from researchers in the area ofcomplex networks.


Financial Networks

V. Boginski, S. Butenko, and P. M. Pardalos, 2005. StatisticalAnalysis of Financial Networks. Computational Statistics andData Analysis 48(2), 431443.

V. Boginski, S. Butenko, and P. M. Pardalos, 2003. OnStructural Properties of the Market Graph. In Innovations inFinancial and Economic Networks, A. Nagurney (ed.), EdwardElgar Publishers, pp. 28-45.

G. A. Bautin, V. A. Kalyagin, A. P. Koldanov, P. A. Koldanov,P. M. Pardalos, 2013. Simple measure of similarity for themarket graph construction, special issue of ComputationalManagement Science on Financial Networks, 10(2-3),105-124..

Recent empirical research has shown that connectionsincrease before and during financial crises.


Empirical Evidence - Jan. 1994 - Dec. 1996

Granger Causality Results: Green Broker, Red Hedge Fund,Black Insurer, Blue Bank Source: Billio, Getmansky, Lo, and Pelizzon (2011)


Empirical Evidence - Jan. 2006 - Dec. 2008

Granger Causality Results: Green Broker, Red Hedge Fund,Black Insurer, Blue Bank Source: Billio, Getmansky, Lo, and Pelizzon (2011)


Financial Networks

Nevertheless, there is very little literature that addresses thevulnerability of financial networks.

Our network performance measure for financialnetworks captures both economic behavior as well asthe underlying network/graph structure and thedynamic reallocation after disruptions.


Financial Networks


The Financial Network Model with Intermediation

This part of the lecture is based on the paper, “Identificationof Critical Nodes and Links in Financial Networks withIntermediation and Electronic Transactions,” A. Nagurney andQ. Qiang, in Computational Methods in Financial Engineering,E. J. Kontoghiorghes, B. Rustem, and P. Winker, Editors,Springer, Berlin, Germany (2008), pp 273-297.



Demand Markets - Uses of Funds

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· · · j · · · � ��n � ��

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Sources of Financial Funds

Intermediaries Non-investment Node

Internet Links

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Physical Links

Physical Links

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Figure: The Structure of the Financial Network with Intermediation



Examples of source agents include households and businesses.

The financial intermediaries, in turn, which can include banks,insurance companies, investment companies, etc., in additionto transacting with the source agents determine how toallocate the incoming financial resources among the distinctuses or financial products associated with the demandmarkets, which correspond to the nodes at the bottom tier ofthe financial network in the figure.

Both source agents and intermediaries maximize their netrevenues while minimizing their risk.

Examples of demand markets are: the markets for real estateloans, household loans, business loans, etc.


The Financial Network Performance Measure

Definition: The Financial Network PerformanceMeasureThe financial network performance measure, EF , for a givennetwork topology G, and demand price functions ρ3k(d)(k = 1, 2, . . . , o), and available funds held by source agents S,is defined as follows:

EF =

∑ok=1

d∗kρ3k (d∗)

o,

where o is the number of demand markets in the financialnetwork, and d∗k and ρ3k(d

∗) denote the equilibrium demandand the equilibrium price for demand market k, respectively.


The Importance of a Financial Network Component

The financial network performance is expected to deterioratewhen a critical network component is eliminated from thenetwork.

Such a component can include a link or a node or a subset ofnodes and links depending on the financial network problemunder investigation. Furthermore, the removal of a criticalnetwork component will cause severe damage than that of thedamage caused by a trivial component.


The Importance of a Financial Network Component

The importance of a network component is defined as:

Definition: Importance of a Financial NetworkComponent

The importance of a financial network component g ∈ G,I (g), is measured by the relative financial networkperformance drop after g is removed from the network:

I (g) =4EF

EF=EF (G )− EF (G − g)

EF (G )

where G − g is the resulting financial network aftercomponent g is removed from network G.


Cybercrime and Financial Institutions

According to a recent survey conducted byPriceWaterhouseCoopers (2012), with over 3,800 respondentsin 78 countries, cybercrime is placing heavy strains on theglobal financial sector, with cybercrime now the second mostcommonly reported economic crime affecting financial servicesfirms.

Cybercrime accounted for 38% of all economic crimesin the financial sector, as compared to an average of16% across all other industries.

The Ponemon Institute (2011) reports that the medianannualized cost of cybercrimes to 50 organizations in its studywas $5.9 million a year, with a range of $1.5 million to $36.5million per company.













Cyber attacks are intrusive and economically costly. Inaddition, they may adversely affect a companys mostvaluable asset its reputation.


Network Economics of Cybercrime

Green Nodes representInstitutionsRed Nodes the AttackersRed Edges betweenAttackers can representcollusion or transactions ofstolen goods.Black Edges betweenInstitutions can showsharing of information andmutual dependence.Blue Edges between theAttacker and Institution canrepresent threats andattacks.


Summary and Conclusions

• We argued for a new Internet – ChoiceNet.

• We developed a game theory model for a service-orientedInternet. The motivation for the research stems, in part, froma need to understand the underlying economics of aservice-oriented Internet with more choices as well as todemonstrate the integration of complex competitive behaviorson multitiered networks.



• We argued for a new Internet – ChoiceNet.

• We developed a game theory model for a service-orientedInternet. The motivation for the research stems, in part, froma need to understand the underlying economics of aservice-oriented Internet with more choices as well as todemonstrate the integration of complex competitive behaviorson multitiered networks.



• We developed both static and dynamic versions of theCournot-Nash-Bertrand game theory model in which theservice providers offer differentiated, but substitutable, servicesand the network providers transport the services to consumersat the demand markets.

• Consumers respond to the composition of service andnetwork provision choices and to the quality levels and servicevolumes, through the prices.

• The service providers compete in a Cournot-Nash manner,whereas the network providers compete a la Bertrand in pricescharged for the transport of the services, as well as with thequality levels associated with the transport.



• We derived the governing equilibrium conditions of theintegrated game theory model and showed that it satisfies avariational inequality problem. We then described theunderlying dynamics, using the theory of projected dynamicalsystems.

• An algorithm was presented, along with convergence results,which provides a discrete-time version of the continuous-timeadjustment processes for the service volumes, quality levels,and prices. We demonstrated the generality of the modelingand computational framework with several numerical examples.

• We also highlighted some of our related research on financialnetworks.












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Topic 7: Future Internet Architectures...SCH-MGMT 825 Management Science Seminar Variational Inequalities, Networks, and Game Theory Spring 2014 ... next-generation Internet. Professor